Lattices in semi-simple Lie groups and multipliers of group C*-algebras

Let Gamma be a lattice in a non-compact simple Lie group G. We prove that the canonical map from the full C*-algebra C*(Gamma) to the multiplier algebra M(C*(G)) is not injective in general (it is never injective if G has Kazhdan's property (T), and not injective for many lattices either in SO(n, 1) or SU(n, 1)). For a locally compact group G, Fell introduced a property (WF3), stating that for any closed subgroup H of G, the canonical map from C*(H) to M(C*(G)) is injective. We prove that, for an almost connected G, property (WF3) is equivalent to amenability.